Pointwise Spectral Asymptotics out of the Diagonal near Degeneration

Abstract

We establish uniform (with respect to $x$, $y$) semiclassical asymptotics and estimates for the Schwartz kernel $e_h(x,y;\tau )$ of spectral projector for a second order elliptic operator inside domain under microhyperbolicity (but not $\xi$-microhyperbolicity) assumption. While such asymptotics for its restriction to the diagonal $e_h(x,x,\tau )$ and, especially, for its trace ${\mathsf{N}}_h(\tau )=\int e_h(x,x,\tau )dx$ are well-known, the out-of-diagonal asymptotics are much less explored, especially uniform ones. Our main tools: microlocal methods, improved successive approximations and geometric optics methods. Our results would also lead to classical asymptotics of $e_h(x,y,\tau )$ for fixed $h$ (say, $h=1$) and $\tau \to \infty$.

arXiv comment27 pp, 1 fig

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• Pointwise Spectral Asymptotics out of the Diagonal near Boundary

Cited by in corpus (1)

• Pointwise Spectral Asymptotics out of the Diagonal near Boundary